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Sine Rule

HigherHigher tier onlyAQAEdexcelOCR

Practise sine rule with this free Higher GCSE Maths worksheet from Virtus Academy. You'll work through applying the sine rule, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Use the sine rule when you have a side and its opposite angle.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

The sine rule works in any triangle, not just right-angled ones. It states that \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), where each side is paired with the angle opposite it.

That pairing is the whole point. Side \(a\) sits opposite angle \(A\), and identifying those pairs correctly is what makes the rule work.

Use the sine rule whenever you have a matching side and angle pair plus one more piece of information. If instead you have two sides and the angle between them, the cosine rule is needed rather than this one.

Revision notes

The rule and when to use it

\(\frac{a}{\sin A} = \frac{b}{\sin B}\). Use it when you have a complete side-angle pair plus one other side or angle.

For finding an angle, it is easier to invert: \(\frac{\sin A}{a} = \frac{\sin B}{b}\).

Finding a side

Put the unknown side on top and rearrange by multiplying.

If \(A = 40^\circ\), \(a = 8\)cm and \(B = 65^\circ\), then \(b = \frac{8 \sin 65}{\sin 40} = 11.3\)cm to one decimal place.

Finding an angle

Use the inverted form, then apply the inverse sine.

With \(a = 7\), \(A = 35^\circ\) and \(b = 10\): \(\sin B = \frac{10 \sin 35}{7}\), giving \(B = 55.0^\circ\).

Key points

  • The sine rule works in any triangle.
  • \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).
  • Each side pairs with the angle opposite it.
  • Use it when you have a complete side-angle pair.
  • Invert the rule when finding an angle.
  • Use the cosine rule instead for two sides and the included angle.

Worked examples

Example 1

In a triangle, \(A = 50^\circ\), \(a = 9\)cm and \(B = 70^\circ\). Find \(b\) to 1 d.p.

Working

\[\frac{b}{\sin 70} = \frac{9}{\sin 50}\]set up the sine rule with b on top
\[b = \frac{9 \sin 70}{\sin 50}\]rearrange to make b the subject
\[= 11.0\text{cm}\]evaluate to one decimal place

Example 2

In a triangle, \(a = 6\)cm, \(A = 40^\circ\) and \(b = 8\)cm. Find \(B\) to 1 d.p.

Working

\[\frac{\sin B}{8} = \frac{\sin 40}{6}\]invert the rule to find an angle
\[\sin B = \frac{8 \sin 40}{6}\]rearrange for sin B
\[B = 59.0^\circ\]apply the inverse sine

Example 3

Explain when the sine rule should be used rather than the cosine rule.

Working

\[\text{You need a matching side and angle pair}\]the sine rule pairs sides with opposite angles
\[\text{Otherwise use the cosine rule}\]two sides and the included angle needs the cosine rule

Common mistakes

  • Pairing a side with the wrong angle.

    Side a must go with angle A, the angle directly opposite it.

  • Using the sine rule without a complete pair.

    If you have two sides and the angle between them, use the cosine rule.

  • Forgetting the inverse sine when finding an angle.

    sin B = 0.85 needs sin⁻¹ to give B.

  • Rounding too early.

    Keep full accuracy and round only the final answer.

Exam tips

  • Label the sides and opposite angles before starting.
  • Put the unknown on top when finding a side.
  • Invert the rule when finding an angle.
  • Check whether a complete side-angle pair exists before choosing the rule.

Key terms

Sine rule
A rule relating sides to the sines of opposite angles.
Opposite angle
The angle facing a given side.
Inverse sine
The function giving an angle from a sine value.
Included angle
The angle between two known sides.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.